#193160

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,7,3,5,9,6,11)(4,12,13,8,14,10,15,16)(17,18,22,27,23,28,21,26)(19,29,32,24,33,39,31,40)(20,34,35,25,37,41,36,42)(30,44,45,43,46,38,47,48);(1,3,7,9,5,11,6,2)(4,10,13,16,14,12,15,8)(17,19,22,32,23,33,21,31)(18,24,27,39,28,40,26,29)(20,30,35,45,37,46,36,47)(25,38,41,48,42,44,34,43);(1,4)(2,8)(3,10)(5,14)(6,15)(7,13)(9,16)(11,12)(17,20,23,37)(18,25,28,42)(19,30,33,46)(21,36,22,35)(24,38,40,44)(26,34,27,41)(29,43,39,48)(31,47,32,45);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,40,39)(25,41,42,34)(30,45,46,47)(38,48,44,43)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100356);